Some more lemmas from CerCo.

[helm.git] / matita / matita / lib / basics / types.ma

diff --git a/matita/matita/lib/basics/types.ma b/matita/matita/lib/basics/types.ma
index a6749bb3991ac36427a53f4b0c04dd49706cf7ae..46cc5dd4be8c0c1a1d25374fa7c2fcdc972d2369 100644 (file)
--- a/matita/matita/lib/basics/types.ma
+++ b/matita/matita/lib/basics/types.ma
@@ -55,19 +55,30 @@ lemma refute_none_by_refl : ∀A,B:Type[0]. ∀P:A → B. ∀Q:B → Type[0]. 
 | #a #H #p normalize @p @refl
 ] qed.
 
+(* dependent pair *)
+record DPair (A:Type[0]) (f:A→Type[0]) : Type[0] ≝ {
+    dpi1:> A
+  ; dpi2: f dpi1
+  }.
+
+interpretation "DPair" 'dpair x = (DPair ? x).
+
+interpretation "mk_DPair" 'mk_DPair x y = (mk_DPair ?? x y).
+
 (* sigma *)
-record Sig (A:Type[0]) (f:A→Type[0]) : Type[0] ≝ {
+record Sig (A:Type[0]) (f:A→Prop) : Type[0] ≝ {
     pi1: A
   ; pi2: f pi1
   }.
   
 interpretation "Sigma" 'sigma x = (Sig ? x).
 
-notation "hvbox(« term 19 a, break term 19 b»)" 
-with precedence 90 for @{ 'dp $a $b }.
-
 interpretation "mk_Sig" 'dp x y = (mk_Sig ?? x y).
 
+lemma sub_pi2 : ∀A.∀P,P':A → Prop. (∀x.P x → P' x) → ∀x:Σx:A.P x. P' (pi1 … x).
+#A #P #P' #H1 * #x #H2 @H1 @H2
+qed.
+
 (* Prod *)
 
 record Prod (A,B:Type[0]) : Type[0] ≝ {
@@ -123,12 +134,6 @@ notation > "hvbox('let' 〈ident x,ident y〉 'as' ident E ≝ t 'in' s)"
 for @{ match $t return λx.x = $t → ? with [ mk_Prod ${ident x} ${ident y} ⇒
         λ${ident E}.$s ] (refl ? $t) }.
 
-(* Prop sigma *)
-record PSig (A:Type[0]) (P:A→Prop) : Type[0] ≝
-  {elem:>A; eproof: P elem}.
-  
-interpretation "subset type" 'sigma x = (PSig ? x).
-
 notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y〉 \nbsp 'as'\nbsp ident E\nbsp ≝ break t \nbsp 'in' \nbsp) break s)"
  with precedence 10
 for @{ match $t return λ${ident k}:$X.$eq $T $k $t → ? with [ mk_Prod (${ident x}:$U) (${ident y}:$W) ⇒